On a Conjecture of Daniel H. Gottlieb
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چکیده
We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible. 1 A version of Gottlieb’s conjecture Let X and Y be finite CW-complexes. In [4, 3], Gottlieb defines a notion of degree of a continuous map f : X → Y as follows. Let f∗ : H∗(X,Z) → H∗(Y,Z) be the induced map in reduced integral homology. The degree deg(f) of f is the least integer n ∈ N, such that there exists a group homomorphism τ : H∗(Y,Z) → H∗(X,Z) which satisfies f∗ ◦ τ = n · id. He conjectures the following (compare [3]): Conjecture 1 (Gottlieb). Let (Y, y) be a finite aspherical CW-complex which is not acyclic. Let f : (X,x) → (Y, y) be a continuous map with deg(f) 6= 0. If χ(Y ) 6= 0, then the centralizer of f∗(π1(X,x)) in π1(Y, y) is trivial. In this note we give a counterexample to this form of the conjecture (see Example 12) and prove a version with a stronger hypothesis, see Theorem 4. Let us rephrase one important consequence of non-vanishing degree in the case of mappings between closed oriented manifolds, so that it is applicable in a more general setting. Definition 2. Let f : (X,x) → (Y, y) be a continuous map. We say that f is a superposition, if for any Qπ1(Y, y)-module L, the induced map f∗ : H π1(X,x) ∗ (X̃ ; f ∗L) → H π1(Y,y) ∗ (Ỹ ;L)
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We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation...
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تاریخ انتشار 2008